LMFDB - Embedded newform 945.2.z.a.314.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(314,945)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("945.314");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.z (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.54586299101$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.31116960000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 $ x^8 + x^6 - 8x^4 + 9x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 315)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			314.1
Root			$0.306808 - 1.70466i$ of defining polynomial
Character	$\chi$	$=$	945.314
Dual form			945.2.z.a.629.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(-1.32288 - 2.29129i) q^{7} +O(q^{10})$	$q+(1.00000 - 1.73205i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(-1.32288 - 2.29129i) q^{7} +(0.311738 - 0.179982i) q^{11} +(-3.56618 + 6.17680i) q^{13} +(-2.00000 - 3.46410i) q^{16} -5.75583i q^{17} +(-3.87298 + 2.23607i) q^{20} +(2.50000 + 4.33013i) q^{25} -5.29150 q^{28} +(-5.12348 + 2.95804i) q^{29} +5.91608i q^{35} -0.719927i q^{44} +(-10.7942 + 6.23202i) q^{47} +(-3.50000 + 6.06218i) q^{49} +(7.13235 + 12.3536i) q^{52} -0.804903 q^{55} -8.00000 q^{64} +(13.8117 - 7.97421i) q^{65} +(-9.96939 - 5.75583i) q^{68} -16.3084i q^{71} -6.32745 q^{73} +(-0.824780 - 0.476187i) q^{77} +(-7.93521 - 13.7442i) q^{79} +8.94427i q^{80} +(15.7789 - 9.10993i) q^{83} +(-6.43521 + 11.1461i) q^{85} +18.8704 q^{91} +(-7.53480 - 13.0507i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4}+O(q^{10}) $ 8 * q + 8 * q^4	$ 8 q + 8 q^{4} - 18 q^{11} - 16 q^{16} + 20 q^{25} - 28 q^{49} - 64 q^{64} + 90 q^{65} - 2 q^{79} + 10 q^{85} + 28 q^{91}+O(q^{100}) $ 8 * q + 8 * q^4 - 18 * q^11 - 16 * q^16 + 20 * q^25 - 28 * q^49 - 64 * q^64 + 90 * q^65 - 2 * q^79 + 10 * q^85 + 28 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/945\mathbb{Z}\right)^\times$.

$n$	$136$	$596$	$757$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$		0			0
$3$		0			0
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$	−1.93649	−	1.11803i	−0.866025	−	0.500000i
$5$	−1.93649	−	1.11803i	−0.866025	−	0.500000i
$6$		0			0
$7$	−1.32288	−	2.29129i	−0.500000	−	0.866025i
$7$	−1.32288	−	2.29129i	−0.500000	−	0.866025i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.311738	−	0.179982i	0.0939925	−	0.0542666i	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	0.311738	−	0.179982i	0.0939925	−	0.0542666i	0.546259	+	0.837616i	$0.316051\pi$
$12$		0			0
$13$	−3.56618	+	6.17680i	−0.989079	+	1.71314i	−0.366900	+	0.930261i	$0.619581\pi$
$13$	−3.56618	+	6.17680i	−0.989079	+	1.71314i	−0.622179	+	0.782875i	$0.713753\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$		−	5.75583i		−	1.39599i	−0.716101	−	0.697997i	$-0.754075\pi$
$17$		−	5.75583i		−	1.39599i	0.716101	−	0.697997i	$-0.245925\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−3.87298	+	2.23607i	−0.866025	+	0.500000i
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$	−5.29150			−1.00000
$29$	−5.12348	+	2.95804i	−0.951405	+	0.549294i	−0.893517	−	0.449029i	$-0.851770\pi$
$29$	−5.12348	+	2.95804i	−0.951405	+	0.549294i	−0.0578882	+	0.998323i	$0.518437\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			5.91608i			1.00000i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$		−	0.719927i		−	0.108533i
$45$		0			0
$46$		0			0
$47$	−10.7942	+	6.23202i	−1.57449	+	0.909033i	−0.578884	+	0.815410i	$0.696511\pi$
$47$	−10.7942	+	6.23202i	−1.57449	+	0.909033i	−0.995608	+	0.0936230i	$0.970155\pi$
$48$		0			0
$49$	−3.50000	+	6.06218i	−0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$	7.13235	+	12.3536i	0.989079	+	1.71314i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−0.804903			−0.108533
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	13.8117	−	7.97421i	1.71314	−	0.989079i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	−9.96939	−	5.75583i	−1.20897	−	0.697997i
$69$		0			0
$70$		0			0
$71$		−	16.3084i		−	1.93545i	−0.252010	−	0.967725i	$-0.581092\pi$
$71$		−	16.3084i		−	1.93545i	0.252010	−	0.967725i	$-0.418908\pi$
$72$		0			0
$73$	−6.32745			−0.740572			−0.370286	−	0.928918i	$-0.620740\pi$
$73$	−6.32745			−0.740572			−0.370286	+	0.928918i	$0.620740\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.824780	−	0.476187i	−0.0939925	−	0.0542666i
$78$		0			0
$79$	−7.93521	−	13.7442i	−0.892781	−	1.54634i	−0.836527	−	0.547926i	$-0.815418\pi$
$79$	−7.93521	−	13.7442i	−0.892781	−	1.54634i	−0.0562544	−	0.998416i	$-0.517916\pi$
$80$			8.94427i			1.00000i
$81$		0			0
$82$		0			0
$83$	15.7789	−	9.10993i	1.73196	−	0.999945i	0.860729	−	0.509064i	$-0.170008\pi$
$83$	15.7789	−	9.10993i	1.73196	−	0.999945i	0.871227	−	0.490881i	$-0.163325\pi$
$84$		0			0
$85$	−6.43521	+	11.1461i	−0.697997	+	1.20897i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	18.8704			1.97816
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−7.53480	−	13.0507i	−0.765043	−	1.32509i	−0.940224	−	0.340557i	$-0.889384\pi$
$97$	−7.53480	−	13.0507i	−0.765043	−	1.32509i	0.175180	−	0.984536i	$-0.443949\pi$
$98$		0			0
$99$		0			0
$100$	10.0000			1.00000
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	1.32288	−	2.29129i	0.130347	−	0.225767i	−0.793463	−	0.608618i	$-0.791724\pi$
$103$	1.32288	−	2.29129i	0.130347	−	0.225767i	0.923810	+	0.382851i	$0.125058\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	9.87043			0.945415			0.472708	−	0.881219i	$-0.343277\pi$
$109$	9.87043			0.945415			0.472708	+	0.881219i	$0.343277\pi$
$110$		0			0
$111$		0			0
$112$	−5.29150	+	9.16515i	−0.500000	+	0.866025i
$113$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$113$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$114$		0			0
$115$		0			0
$116$			11.8322i			1.09859i
$117$		0			0
$118$		0			0
$119$	−13.1883	+	7.61425i	−1.20897	+	0.697997i
$120$		0			0
$121$	−5.43521	+	9.41407i	−0.494110	+	0.855824i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	1.00000i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$	10.2470	+	5.91608i	0.866025	+	0.500000i
$141$		0			0
$142$		0			0
$143$			2.56739i			0.214696i
$144$		0			0
$145$	13.2288			1.09859
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−14.7470	−	8.51416i	−1.20812	−	0.697507i	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−14.7470	−	8.51416i	−1.20812	−	0.697507i	−0.962348	+	0.271821i	$0.912374\pi$
$150$		0			0
$151$	−3.43521	−	5.94996i	−0.279554	−	0.484201i	0.691720	−	0.722166i	$-0.256853\pi$
$151$	−3.43521	−	5.94996i	−0.279554	−	0.484201i	−0.971274	+	0.237964i	$0.923520\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	11.1010	−	19.2275i	0.885954	−	1.53452i	0.0413387	−	0.999145i	$-0.486838\pi$
$157$	11.1010	−	19.2275i	0.885954	−	1.53452i	0.844616	−	0.535373i	$-0.179829\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	16.6036	+	9.58612i	1.28483	+	0.741796i	0.977727	−	0.209881i	$-0.0673075\pi$
$167$	16.6036	+	9.58612i	1.28483	+	0.741796i	0.307102	+	0.951677i	$0.400641\pi$
$168$		0			0
$169$	−18.9352	−	32.7968i	−1.45655	−	2.52283i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.68246	−	5.59017i	0.736144	−	0.425013i	−0.0845218	−	0.996422i	$-0.526936\pi$
$173$	9.68246	−	5.59017i	0.736144	−	0.425013i	0.820666	+	0.571409i	$0.193603\pi$
$174$		0			0
$175$	6.61438	−	11.4564i	0.500000	−	0.866025i
$176$	−1.24695	−	0.719927i	−0.0939925	−	0.0542666i
$177$		0			0
$178$		0			0
$179$			14.8685i			1.11133i	0.831408	+	0.555663i	$0.187536\pi$
$179$			14.8685i			1.11133i	−0.831408	+	0.555663i	$0.812464\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.03594	−	1.79431i	−0.0757558	−	0.131213i
$188$			24.9281i			1.81807i
$189$		0			0
$190$		0			0
$191$	−5.12348	+	2.95804i	−0.370722	+	0.214036i	−0.673774	−	0.738938i	$-0.735328\pi$
$191$	−5.12348	+	2.95804i	−0.370722	+	0.214036i	0.303052	+	0.952974i	$0.401994\pi$
$192$		0			0
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$	7.00000	+	12.1244i	0.500000	+	0.866025i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.5554	+	7.82624i	0.951405	+	0.549294i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	28.5294			1.97816
$209$		0			0
$210$		0			0
$211$	−1.93521	+	3.35189i	−0.133226	+	0.230753i	−0.924918	−	0.380166i	$-0.875867\pi$
$211$	−1.93521	+	3.35189i	−0.133226	+	0.230753i	0.791693	+	0.610920i	$0.209200\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.804903	+	1.39413i	−0.0542666	+	0.0939925i
$221$	35.5526	+	20.5263i	2.39153	+	1.38075i
$222$		0			0
$223$	4.37108	+	7.57093i	0.292709	+	0.506987i	0.974449	−	0.224607i	$-0.0721099\pi$
$223$	4.37108	+	7.57093i	0.292709	+	0.506987i	−0.681740	+	0.731594i	$0.738777\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.63426	−	3.83029i	0.440331	−	0.254225i	−0.263407	−	0.964685i	$-0.584846\pi$
$227$	6.63426	−	3.83029i	0.440331	−	0.254225i	0.703738	+	0.710460i	$0.251513\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	27.8704			1.81807
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−25.6174	−	14.7902i	−1.65705	−	0.956698i	−0.974066	−	0.226266i	$-0.927348\pi$
$239$	−25.6174	−	14.7902i	−1.65705	−	0.956698i	−0.682985	−	0.730433i	$-0.739318\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	13.5554	−	7.82624i	0.866025	−	0.500000i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	25.7483	+	14.8658i	1.60613	+	0.927301i	0.990225	+	0.139482i	$0.0445438\pi$
$257$	25.7483	+	14.8658i	1.60613	+	0.927301i	0.615907	+	0.787819i	$0.288790\pi$
$258$		0			0
$259$		0			0
$260$		−	31.8968i		−	1.97816i
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−19.9388	+	11.5117i	−1.20897	+	0.697997i
$273$		0			0
$274$		0			0
$275$	1.55869	+	0.899909i	0.0939925	+	0.0542666i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	27.3117	−	15.7684i	1.62928	−	0.940666i	0.644974	−	0.764204i	$-0.276868\pi$
$281$	27.3117	−	15.7684i	1.62928	−	0.940666i	0.984307	−	0.176462i	$-0.0564652\pi$
$282$		0			0
$283$	13.8622	−	24.0101i	0.824025	−	1.42725i	−0.0786368	−	0.996903i	$-0.525057\pi$
$283$	13.8622	−	24.0101i	0.824025	−	1.42725i	0.902662	−	0.430350i	$-0.141610\pi$
$284$	−28.2470	−	16.3084i	−1.67615	−	0.967725i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.1296			−0.948798
$290$		0			0
$291$		0			0
$292$	−6.32745	+	10.9595i	−0.370286	+	0.641354i
$293$	−21.3014	−	12.2984i	−1.24444	−	0.718479i	−0.274446	−	0.961602i	$-0.588495\pi$
$293$	−21.3014	−	12.2984i	−1.24444	−	0.718479i	−0.969995	+	0.243124i	$0.921828\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	23.0069			1.31307			0.656535	−	0.754295i	$-0.272021\pi$
$307$	23.0069			1.31307			0.656535	+	0.754295i	$0.272021\pi$
$308$	−1.64956	+	0.952374i	−0.0939925	+	0.0542666i
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−14.5516	−	25.2042i	−0.822507	−	1.42462i	−0.903810	−	0.427934i	$-0.859241\pi$
$313$	−14.5516	−	25.2042i	−0.822507	−	1.42462i	0.0813030	−	0.996689i	$-0.474092\pi$
$314$		0			0
$315$		0			0
$316$	−31.7409			−1.78556
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$	−1.06479	+	1.84427i	−0.0596166	+	0.103259i
$320$	15.4919	+	8.94427i	0.866025	+	0.500000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−35.6618			−1.97816
$326$		0			0
$327$		0			0
$328$		0			0
$329$	28.5587	+	16.4884i	1.57449	+	0.909033i
$330$		0			0
$331$	17.3704	+	30.0865i	0.954765	+	1.65370i	0.734905	+	0.678170i	$0.237227\pi$
$331$	17.3704	+	30.0865i	0.954765	+	1.65370i	0.219860	+	0.975531i	$0.429440\pi$
$332$		−	36.4397i		−	1.99989i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$337$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$338$		0			0
$339$		0			0
$340$	12.8704	+	22.2922i	0.697997	+	1.20897i
$341$		0			0
$342$		0			0
$343$	18.5203			1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−25.1744	+	14.5344i	−1.33990	+	0.773590i	−0.986792	−	0.161993i	$-0.948208\pi$
$353$	−25.1744	+	14.5344i	−1.33990	+	0.773590i	−0.353106	+	0.935583i	$0.614874\pi$
$354$		0			0
$355$	−18.2333	+	31.5811i	−0.967725	+	1.67615i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			11.8322i			0.624477i	0.950004	+	0.312239i	$0.101079\pi$
$359$			11.8322i			0.624477i	−0.950004	+	0.312239i	$0.898921\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$		0			0
$364$	18.8704	−	32.6845i	0.989079	−	1.71314i
$365$	12.2530	+	7.07430i	0.641354	+	0.370286i
$366$		0			0
$367$	9.89362	+	17.1363i	0.516443	+	0.894505i	0.999818	+	0.0190919i	$0.00607750\pi$
$367$	9.89362	+	17.1363i	0.516443	+	0.894505i	−0.483375	+	0.875413i	$0.660589\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	42.1956i		−	2.17318i
$378$		0			0
$379$	−22.7409			−1.16812			−0.584060	−	0.811711i	$-0.698537\pi$
$379$	−22.7409			−1.16812			−0.584060	+	0.811711i	$0.698537\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−27.3978	−	15.8181i	−1.39996	−	0.808269i	−0.405575	−	0.914062i	$-0.632929\pi$
$383$	−27.3978	−	15.8181i	−1.39996	−	0.808269i	−0.994388	+	0.105793i	$0.966262\pi$
$384$		0			0
$385$	1.06479	+	1.84427i	0.0542666	+	0.0939925i
$386$		0			0
$387$		0			0
$388$	−30.1392			−1.53009
$389$	−26.6883	+	15.4085i	−1.35315	+	0.781241i	−0.988689	−	0.149979i	$-0.952080\pi$
$389$	−26.6883	+	15.4085i	−1.35315	+	0.781241i	−0.364459	+	0.931219i	$0.618746\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$			35.4874i			1.78556i
$396$		0			0
$397$	−34.3948			−1.72622			−0.863112	−	0.505013i	$-0.831488\pi$
$397$	−34.3948			−1.72622			−0.863112	+	0.505013i	$0.831488\pi$
$398$		0			0
$399$		0			0
$400$	10.0000	−	17.3205i	0.500000	−	0.866025i
$401$	−25.6174	−	14.7902i	−1.27927	−	0.738587i	−0.302556	−	0.953131i	$-0.597840\pi$
$401$	−25.6174	−	14.7902i	−1.27927	−	0.738587i	−0.976714	+	0.214544i	$0.931173\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$	−2.64575	−	4.58258i	−0.130347	−	0.225767i
$413$		0			0
$414$		0			0
$415$	−40.7409			−1.99989
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	−16.9352	−	29.3326i	−0.825372	−	1.42959i	−0.901635	−	0.432498i	$-0.857632\pi$
$421$	−16.9352	−	29.3326i	−0.825372	−	1.42959i	0.0762630	−	0.997088i	$-0.475701\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	24.9235	−	14.3896i	1.20897	−	0.697997i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$			18.1082i			0.872241i	0.899888	+	0.436121i	$0.143648\pi$
$431$			18.1082i			0.872241i	−0.899888	+	0.436121i	$0.856352\pi$
$432$		0			0
$433$	−10.5830			−0.508587			−0.254293	−	0.967127i	$-0.581843\pi$
$433$	−10.5830			−0.508587			−0.254293	+	0.967127i	$0.581843\pi$
$434$		0			0
$435$		0			0
$436$	9.87043	−	17.0961i	0.472708	−	0.818754i
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$443$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	10.5830	+	18.3303i	0.500000	+	0.866025i
$449$		−	41.4126i		−	1.95438i	−0.212368	−	0.977190i	$-0.568118\pi$
$449$		−	41.4126i		−	1.95438i	0.212368	−	0.977190i	$-0.431882\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−36.5424	−	21.0978i	−1.71314	−	0.989079i
$456$		0			0
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$	20.4939	+	11.8322i	0.951405	+	0.549294i
$465$		0			0
$466$		0			0
$467$			14.3688i			0.664908i	0.943119	+	0.332454i	$0.107877\pi$
$467$			14.3688i			0.664908i	−0.943119	+	0.332454i	$0.892123\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$			30.4570i			1.39599i
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	10.8704	+	18.8281i	0.494110	+	0.855824i
$485$			33.6967i			1.53009i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11.9413	−	6.89432i	−0.538904	−	0.311136i	0.205731	−	0.978609i	$-0.434043\pi$
$491$	−11.9413	−	6.89432i	−0.538904	−	0.311136i	−0.744635	+	0.667472i	$0.767376\pi$
$492$		0			0
$493$	17.0260	+	29.4899i	0.766811	+	1.32816i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−37.3672	+	21.5740i	−1.67615	+	0.967725i
$498$		0			0
$499$	2.56479	−	4.44234i	0.114816	−	0.198867i	−0.802890	−	0.596127i	$-0.796706\pi$
$499$	2.56479	−	4.44234i	0.114816	−	0.198867i	0.917706	+	0.397260i	$0.130039\pi$
$500$	−19.3649	−	11.1803i	−0.866025	−	0.500000i
$501$		0			0
$502$		0			0
$503$			38.0132i			1.69492i	0.530857	+	0.847461i	$0.321870\pi$
$503$			38.0132i			1.69492i	−0.530857	+	0.847461i	$0.678130\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	8.37043	+	14.4980i	0.370286	+	0.641354i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.12348	+	2.95804i	−0.225767	+	0.130347i
$516$		0			0
$517$	−2.24330	+	3.88551i	−0.0986602	+	0.170884i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	4.71764			0.206288			0.103144	−	0.994666i	$-0.467110\pi$
$523$	4.71764			0.206288			0.103144	+	0.994666i	$0.467110\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			2.51975i			0.108533i
$540$		0			0
$541$	36.8704			1.58518			0.792592	−	0.609753i	$-0.208731\pi$
$541$	36.8704			1.58518			0.792592	+	0.609753i	$0.208731\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−19.1140	−	11.0355i	−0.818754	−	0.472708i
$546$		0			0
$547$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$547$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−20.9946	+	36.3637i	−0.892781	+	1.54634i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$	20.4939	−	11.8322i	0.866025	−	0.500000i
$561$		0			0
$562$		0			0
$563$	7.45904	+	4.30648i	0.314361	+	0.181496i	0.648876	−	0.760894i	$-0.275239\pi$
$563$	7.45904	+	4.30648i	0.314361	+	0.181496i	−0.334515	+	0.942390i	$0.608573\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15.9939	+	9.23408i	−0.670499	+	0.387113i	−0.796266	−	0.604947i	$-0.793194\pi$
$569$	−15.9939	+	9.23408i	−0.670499	+	0.387113i	0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$	23.3704	−	40.4788i	0.978022	−	1.69398i	0.308443	−	0.951243i	$-0.400192\pi$
$571$	23.3704	−	40.4788i	0.978022	−	1.69398i	0.669579	−	0.742741i	$-0.266474\pi$
$572$	4.44685	+	2.56739i	0.185932	+	0.107348i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−11.8500			−0.493322			−0.246661	−	0.969102i	$-0.579333\pi$
$577$	−11.8500			−0.493322			−0.246661	+	0.969102i	$0.579333\pi$
$578$		0			0
$579$		0			0
$580$	13.2288	−	22.9129i	0.549294	−	0.951405i
$581$	−41.7470	−	24.1026i	−1.73196	−	0.999945i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.74597	+	4.47214i	−0.319710	+	0.184585i	−0.651263	−	0.758852i	$-0.725761\pi$
$587$	−7.74597	+	4.47214i	−0.319710	+	0.184585i	0.331553	+	0.943437i	$0.392427\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	42.4853i		−	1.74466i	−0.488916	−	0.872331i	$-0.662608\pi$
$593$		−	42.4853i		−	1.74466i	0.488916	−	0.872331i	$-0.337392\pi$
$594$		0			0
$595$	34.0519			1.39599
$596$	−29.4939	+	17.0283i	−1.20812	+	0.697507i
$597$		0			0
$598$		0			0
$599$	42.0587	+	24.2826i	1.71847	+	0.992160i	0.921723	+	0.387849i	$0.126782\pi$
$599$	42.0587	+	24.2826i	1.71847	+	0.992160i	0.796748	+	0.604311i	$0.206552\pi$
$600$		0			0
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$		0			0
$604$	−13.7409			−0.559107
$605$	21.0505	−	12.1535i	0.855824	−	0.494110i
$606$		0			0
$607$	−22.4889	+	38.9519i	−0.912796	+	1.58101i	−0.102699	+	0.994712i	$0.532748\pi$
$607$	−22.4889	+	38.9519i	−0.912796	+	1.58101i	−0.810097	+	0.586296i	$0.800586\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	88.8979i		−	3.59642i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$617$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	−22.2020	−	38.4549i	−0.885954	−	1.53452i
$629$		0			0
$630$		0			0
$631$	−8.12957			−0.323633			−0.161817	−	0.986821i	$-0.551735\pi$
$631$	−8.12957			−0.323633			−0.161817	+	0.986821i	$0.551735\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−24.9632	−	43.2376i	−0.989079	−	1.71314i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	40.9878	−	23.6643i	1.61892	−	0.934684i	0.631721	−	0.775196i	$-0.282349\pi$
$641$	40.9878	−	23.6643i	1.61892	−	0.934684i	0.987200	−	0.159489i	$-0.0509845\pi$
$642$		0			0
$643$	−15.4721	+	26.7984i	−0.610158	+	1.05683i	0.381055	+	0.924552i	$0.375561\pi$
$643$	−15.4721	+	26.7984i	−0.610158	+	1.05683i	−0.991213	+	0.132273i	$0.957772\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			17.8885i			0.703271i	0.936137	+	0.351636i	$0.114374\pi$
$647$			17.8885i			0.703271i	−0.936137	+	0.351636i	$0.885626\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	40.8117	−	23.5627i	1.58980	−	0.917871i	0.596461	−	0.802642i	$-0.296573\pi$
$659$	40.8117	−	23.5627i	1.58980	−	0.917871i	0.993339	−	0.115229i	$-0.0367601\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	33.2073	−	19.1722i	1.28483	−	0.741796i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$		0			0
$675$		0			0
$676$	−75.7409			−2.91311
$677$	−28.2226	+	16.2943i	−1.08468	+	0.626242i	−0.932156	−	0.362058i	$-0.882074\pi$
$677$	−28.2226	+	16.2943i	−1.08468	+	0.626242i	−0.152527	+	0.988299i	$0.548741\pi$
$678$		0			0
$679$	−19.9352	+	34.5288i	−0.765043	+	1.32509i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$		−	22.3607i		−	0.850026i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−13.2288	−	22.9129i	−0.500000	−	0.866025i
$701$			49.2851i			1.86147i	0.365690	+	0.930737i	$0.380833\pi$
$701$			49.2851i			1.86147i	−0.365690	+	0.930737i	$0.619167\pi$
$702$		0			0
$703$		0			0
$704$	−2.49390	+	1.43985i	−0.0939925	+	0.0542666i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.500000	+	0.866025i	0.0187779	+	0.0325243i	0.875262	−	0.483650i	$-0.160689\pi$
$709$	0.500000	+	0.866025i	0.0187779	+	0.0325243i	−0.856484	+	0.516174i	$0.827356\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	2.87043	−	4.97172i	0.107348	−	0.185932i
$716$	25.7530	+	14.8685i	0.962437	+	0.555663i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−7.00000			−0.260694
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−25.6174	−	14.7902i	−0.951405	−	0.549294i
$726$		0			0
$727$	24.5608	+	42.5405i	0.910909	+	1.57774i	0.812783	+	0.582566i	$0.197951\pi$
$727$	24.5608	+	42.5405i	0.910909	+	1.57774i	0.0981255	+	0.995174i	$0.468715\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	25.1346	−	43.5345i	0.928369	−	1.60798i	0.142318	−	0.989821i	$-0.454545\pi$
$733$	25.1346	−	43.5345i	0.928369	−	1.60798i	0.786051	−	0.618161i	$-0.212122\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	11.0000			0.404642			0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000			0.404642			0.202321	+	0.979319i	$0.435152\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$	19.0382	+	32.9752i	0.697507	+	1.20812i
$746$		0			0
$747$		0			0
$748$	−4.14378			−0.151512
$749$		0			0
$750$		0			0
$751$	6.50000	−	11.2583i	0.237188	−	0.410822i	−0.722718	−	0.691143i	$-0.757107\pi$
$751$	6.50000	−	11.2583i	0.237188	−	0.410822i	0.959906	+	0.280321i	$0.0904408\pi$
$752$	43.1767	+	24.9281i	1.57449	+	0.909033i
$753$		0			0
$754$		0			0
$755$			15.3627i			0.559107i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	−13.0573	−	22.6160i	−0.472708	−	0.818754i
$764$			11.8322i			0.428073i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	46.9990i		−	1.69044i	−0.534421	−	0.845218i	$-0.679470\pi$
$773$		−	46.9990i		−	1.69044i	0.534421	−	0.845218i	$-0.320530\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−2.93521	−	5.08394i	−0.105030	−	0.181918i
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$	−42.9939	+	24.8225i	−1.53452	+	0.885954i
$786$		0			0
$787$	25.7681	−	44.6317i	0.918535	−	1.59095i	0.116892	−	0.993145i	$-0.462707\pi$
$787$	25.7681	−	44.6317i	0.918535	−	1.59095i	0.801642	−	0.597804i	$-0.203960\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.2532	−	10.5385i	−0.646562	−	0.373293i	0.140576	−	0.990070i	$-0.455105\pi$
$797$	−18.2532	−	10.5385i	−0.646562	−	0.373293i	−0.787138	+	0.616777i	$0.788438\pi$
$798$		0			0
$799$	35.8704	+	62.1294i	1.26900	+	2.19798i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.97250	+	1.13883i	−0.0696081	+	0.0401883i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	13.0687i		−	0.459471i	−0.973253	−	0.229736i	$-0.926214\pi$
$809$		−	13.0687i		−	0.459471i	0.973253	−	0.229736i	$-0.0737862\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	27.1109	−	15.6525i	0.951405	−	0.549294i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−40.1883	+	23.2027i	−1.40258	+	0.809780i	−0.994657	−	0.103236i	$-0.967080\pi$
$821$	−40.1883	+	23.2027i	−1.40258	+	0.809780i	−0.407923	+	0.913016i	$0.633747\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	28.5294	−	49.4144i	0.989079	−	1.71314i
$833$	34.8929	+	20.1454i	1.20897	+	0.697997i
$834$		0			0
$835$	−21.4352	−	37.1269i	−0.741796	−	1.28483i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	3.00000	−	5.19615i	0.103448	−	0.179178i
$842$		0			0
$843$		0			0
$844$	3.87043	+	6.70377i	0.133226	+	0.230753i
$845$			84.6808i			2.91311i
$846$		0			0
$847$	28.7604			0.988221
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	21.1660	+	36.6606i	0.724710	+	1.25524i	0.959093	+	0.283091i	$0.0913598\pi$
$853$	21.1660	+	36.6606i	0.724710	+	1.25524i	−0.234383	+	0.972144i	$0.575307\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.51035	+	1.44935i	−0.0857520	+	0.0495090i	−0.542263	−	0.840209i	$-0.682432\pi$
$857$	−2.51035	+	1.44935i	−0.0857520	+	0.0495090i	0.456511	+	0.889718i	$0.349099\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−25.0000			−0.850026
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.94741	−	2.85639i	−0.167829	−	0.0968963i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−25.6174	+	14.7902i	−0.866025	+	0.500000i
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$	1.60981	+	2.78827i	0.0542666	+	0.0939925i
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	71.1052	−	41.0526i	2.39153	−	1.38075i
$885$		0			0
$886$		0			0
$887$	30.9839	+	17.8885i	1.04034	+	0.600639i	0.919929	−	0.392086i	$-0.128246\pi$
$887$	30.9839	+	17.8885i	1.04034	+	0.600639i	0.120408	+	0.992725i	$0.461580\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	17.4843			0.585418
$893$		0			0
$894$		0			0
$895$	16.6235	−	28.7928i	0.555663	−	0.962437i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		−	15.3212i		−	0.508450i
$909$		0			0
$910$		0			0
$911$	16.6174	−	9.59405i	0.550558	−	0.317865i	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	16.6174	−	9.59405i	0.550558	−	0.317865i	0.749347	+	0.662177i	$0.230367\pi$
$912$		0			0
$913$	3.27924	−	5.67982i	0.108527	−	0.187975i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	29.0000			0.956622			0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000			0.956622			0.478311	+	0.878191i	$0.341249\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	100.734	+	58.1586i	3.31569	+	1.91431i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$			4.63289i			0.151512i
$936$		0			0
$937$	−24.6167			−0.804191			−0.402096	−	0.915598i	$-0.631718\pi$
$937$	−24.6167			−0.804191			−0.402096	+	0.915598i	$0.631718\pi$
$938$		0			0
$939$		0			0
$940$	27.8704	−	48.2730i	0.909033	−	1.57449i
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$	22.5648	−	39.0834i	0.732484	−	1.26870i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	13.2288			0.428073
$956$	−51.2348	+	29.5804i	−1.65705	+	0.956698i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.5000	−	26.8468i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$		−	31.3050i		−	1.00000i
$981$		0			0
$982$		0			0
$983$	−29.0834	+	16.7913i	−0.927616	+	0.535559i	−0.886057	−	0.463577i	$-0.846566\pi$
$983$	−29.0834	+	16.7913i	−0.927616	+	0.535559i	−0.0415592	+	0.999136i	$0.513233\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−4.74085			−0.150598			−0.0752991	−	0.997161i	$-0.523991\pi$
$991$	−4.74085			−0.150598			−0.0752991	+	0.997161i	$0.523991\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9.26013	+	16.0390i	0.293271	+	0.507961i	0.974581	−	0.224034i	$-0.0719228\pi$
$997$	9.26013	+	16.0390i	0.293271	+	0.507961i	−0.681310	+	0.731995i	$0.738589\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.z.a.314.1		8
3.2	odd	2		315.2.z.a.104.3	yes	8
5.4	even	2	inner	945.2.z.a.314.4		8
7.6	odd	2	inner	945.2.z.a.314.4		8
9.2	odd	6	inner	945.2.z.a.629.1		8
9.7	even	3		315.2.z.a.209.4	yes	8
15.14	odd	2		315.2.z.a.104.2	✓	8
21.20	even	2		315.2.z.a.104.2	✓	8
35.34	odd	2	CM	945.2.z.a.314.1		8
45.29	odd	6	inner	945.2.z.a.629.4		8
45.34	even	6		315.2.z.a.209.1	yes	8
63.20	even	6	inner	945.2.z.a.629.4		8
63.34	odd	6		315.2.z.a.209.1	yes	8
105.104	even	2		315.2.z.a.104.3	yes	8
315.34	odd	6		315.2.z.a.209.4	yes	8
315.209	even	6	inner	945.2.z.a.629.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.z.a.104.2	✓	8	15.14	odd	2
315.2.z.a.104.2	✓	8	21.20	even	2
315.2.z.a.104.3	yes	8	3.2	odd	2
315.2.z.a.104.3	yes	8	105.104	even	2
315.2.z.a.209.1	yes	8	45.34	even	6
315.2.z.a.209.1	yes	8	63.34	odd	6
315.2.z.a.209.4	yes	8	9.7	even	3
315.2.z.a.209.4	yes	8	315.34	odd	6
945.2.z.a.314.1		8	1.1	even	1	trivial
945.2.z.a.314.1		8	35.34	odd	2	CM
945.2.z.a.314.4		8	5.4	even	2	inner
945.2.z.a.314.4		8	7.6	odd	2	inner
945.2.z.a.629.1		8	9.2	odd	6	inner
945.2.z.a.629.1		8	315.209	even	6	inner
945.2.z.a.629.4		8	45.29	odd	6	inner
945.2.z.a.629.4		8	63.20	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.z (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(-1.32288 - 2.29129i) q^{7} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(-1.32288 - 2.29129i) q^{7} +(0.311738 - 0.179982i) q^{11} +(-3.56618 + 6.17680i) q^{13} +(-2.00000 - 3.46410i) q^{16} -5.75583i q^{17} +(-3.87298 + 2.23607i) q^{20} +(2.50000 + 4.33013i) q^{25} -5.29150 q^{28} +(-5.12348 + 2.95804i) q^{29} +5.91608i q^{35} -0.719927i q^{44} +(-10.7942 + 6.23202i) q^{47} +(-3.50000 + 6.06218i) q^{49} +(7.13235 + 12.3536i) q^{52} -0.804903 q^{55} -8.00000 q^{64} +(13.8117 - 7.97421i) q^{65} +(-9.96939 - 5.75583i) q^{68} -16.3084i q^{71} -6.32745 q^{73} +(-0.824780 - 0.476187i) q^{77} +(-7.93521 - 13.7442i) q^{79} +8.94427i q^{80} +(15.7789 - 9.10993i) q^{83} +(-6.43521 + 11.1461i) q^{85} +18.8704 q^{91} +(-7.53480 - 13.0507i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4}+O(q^{10}) \) 8 * q + 8 * q^4	\( 8 q + 8 q^{4} - 18 q^{11} - 16 q^{16} + 20 q^{25} - 28 q^{49} - 64 q^{64} + 90 q^{65} - 2 q^{79} + 10 q^{85} + 28 q^{91}+O(q^{100}) \) 8 * q + 8 * q^4 - 18 * q^11 - 16 * q^16 + 20 * q^25 - 28 * q^49 - 64 * q^64 + 90 * q^65 - 2 * q^79 + 10 * q^85 + 28 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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